Problem 39 - Algebra 2

Question

Evaluate each infinite series that has a sum. $$ \sum_{n=1}^{\infty} 2(1.2)^{n-1} $$

Step-by-Step Solution

Answer

The sum does not exist because the common ratio is greater than 1.

Step 1: Identify the first term and the common ratio

In the series \(\sum_{n=1}^{\infty} 2(1.2)^{n-1}\), \(a = 2\) and \(r = 1.2\).

Step 2: Check if the sum of the series exists

The sum of the series exists if the absolute value of the common ratio \(|r|\) is less than 1. In this case, \(|r| = 1.2\), which is greater than 1, so the sum doesn't exist.

Key Concepts

Geometric Series

In mathematics, a geometric series is a series of numbers with a constant ratio between successive terms. This means that to go from one term to the next, you multiply by the same number each time. For example, in a geometric series like \(2, 4, 8, 16, \ldots\), the common ratio \(r\) is 2 because each term is 2 times the previous term.

A geometric series can be written in the form:

• First term \(a\)
• Common ratio \(r\)
• Terms: \(a, ar, ar^2, ar^3, \ldots\)

The series continues indefinitely, adding up all these terms. It's crucial to notice whether a geometric series converges, which is an important concept when determining the sum.

Convergence

Convergence is a concept that indicates whether a series approaches a specific value as more of its terms are added. For a geometric series, convergence is determined by the value of the common ratio \(r\).

If the common ratio's absolute value \(|r|\) is less than 1, the series converges, meaning it will approach a finite sum as more and more terms are added. If \(|r|\) is greater than or equal to 1, the series diverges, suggesting it does not settle to a finite value.

• Convergent Series: \(|r| < 1\)
• Divergent Series: \(|r| \geq 1\)

In our original exercise, \(|r| = 1.2\) which is greater than 1, making the series divergent. This means it doesn't have a set sum and continues to grow indefinitely.

Sum of Series

The sum of a series is the value that the series approaches as an infinite number of terms are added together. For a convergent geometric series, there is a convenient formula to find the sum:

• Sum \(S = \frac{a}{1 - r}\)

where \(a\) is the first term and \(r\) is the common ratio. It's important to remember that this formula only works if the series is convergent \((|r| < 1)\).

In cases where \(|r|\) is not less than 1, like our original problem, the series does not have a sum because it diverges. The terms keep increasing and do not approach a fixed value. This is why determining the convergence before finding the sum is so crucial in geometric series.